Least-Squares Solutions of Generalized Sylvester-Type Quaternion Matrix Equations

Abstract

This paper focuses on finding solutions of generalized Sylvester-type matrix equations over the quaternion skew-field. We express the general least–squares solutions, and perhermitian, skew-perhermitian least-squares solutions of \(AXB+CYD=E\) and \(AXB+CXD=E\) over the quaternion skew-field in terms of a vec operator (defined specifically for matrices over the quaternion skew-field) and the Moore–Penrose pseudoinverse. In addition, characterizations that facilitate the computation of the least-squares solutions closest to prescribed quaternion matrices are deduced. We illustrate our theoretical findings on several numerical examples, most of which originate from color image restoration via Tikhonov regularization.